Title: Algebraic Aspects of Lattice Simplices

Abstract: Given a lattice polytope P, there are open problems of interest related to the integer decomposition property, Ehrhart h*-unimodality, and Ehrhart positivity. In this talk, we will survey some recent results in this area, based on various joint works with Rob Davis, Morgan Lane, Fu Liu, and Liam Solus.

Title: Algebraic Aspects of Lattice Simplices

Abstract: Given a lattice polytope P, there are open problems of interest related to the integer decomposition property, Ehrhart h*-unimodality, and Ehrhart positivity. In this talk, we will survey some recent results in this area, based on various joint works with Rob Davis, Morgan Lane, Fu Liu, and Liam Solus.

Title: Mutation of friezes

Title: Mutation of friezes

Speaker:  Gabor Hetyei, UNC Charlotte

Title:  Alternation acyclic tournaments and the homogeneous Linial arrangement

Abstract:

We define a tournament to be alternation acyclic if it does not

contain a cycle in which descents and ascents alternate. We show that

these label the regions in a homogenized generalization of the Linial

arrangement. Using a result by Athanasiadis, we show that these are

counted by the median Genocchi numbers. By establishing a bijection

with objects defined by Dumont, we show that alternation acyclic

tournaments in which at least one ascent begins at each vertex, except

for the largest one, are counted by the Genocchi numbers of the first

kind. As an unexpected consequence, we obtain a simple model for the

normalized median Genocchi numbers.

Speaker:  Galen Dorpalen-Barry, University of Minnesota

Title:  Whitney Numbers for Cones

Abstract:

An arrangement of hyperplanes dissects space into connected components

called chambers. A nonempty intersection of halfspaces from the

arrangement will be called a cone.  The number of chambers of the

arrangement lying within the cone is counted by a theorem of

Zaslavsky, as a sum of certain nonnegative integers that we will call

the cone's "Whitney numbers of the 1st kind".  For cones inside the

reflection arrangement of type A (the braid arrangement), cones

correspond to posets, chambers in the cone correspond to linear

extensions of the poset, and these Whitney numbers refine the number

of linear extensions.  We present some basic facts about these Whitney numbers,

and interpret them for two families of posets.

Speaker:  McCabe Olsen, Ohio State University

Title:  Signed Birkhoff polytopes and the orthant-lattice preservation property

Abstract:

Given a d-dimensional lattice polytope P, we say that P has the orthant-lattice preservation property (OLP) if the subpolytope obtained by restriction to any orthant is a lattice polytope. While this property feels somewhat contrived, it can actually be quite useful in verification of discrete geometric properties of P. In this talk, we will discuss a number of results for the existence of triangulation and the integer decomposition property for reflexive OLP polytopes. One such polytope which fits into the program is a type-B analogue of the Birkhoff polytope and its dual polytope, the investigation of which led to interest in this property. This is based on joint work with Florian Kohl (Aalto University).

Speaker:  Susanna Lang, U Kentucky

Title:  Rational Catalan Numbers and Associahedra

Abstract:

Classical Catalan numbers are known to count over 200 combinatorial objects, including Dyck paths, noncrossing partitions, and vertices of the classical associahedra. In this talk we discuss a generalization of the classical Catalan numbers and their connection with a class of simplicial complexes known as rational associahedra. We show rational associahedra have many nice properties, in particular they are shellable. This talk follows the paper "Rational Associahedra and Noncrossing Partitions" by Armstrong, Rhoades, and Williams.

This is Susanna Lang's Masters exam.

Speaker:  Susanna Lang, U Kentucky

Title:  Rational Catalan Numbers and Associahedra

Abstract:

Classical Catalan numbers are known to count over 200 combinatorial objects, including Dyck paths, noncrossing partitions, and vertices of the classical associahedra. In this talk we discuss a generalization of the classical Catalan numbers and their connection with a class of simplicial complexes known as rational associahedra. We show rational associahedra have many nice properties, in particular they are shellable. This talk follows the paper "Rational Associahedra and Noncrossing Partitions" by Armstrong, Rhoades, and Williams.

This is Susanna Lang's Masters exam.